Telecommunication Network Capacity Design for Uncertain Demand

The expansion of telecommunication services has increased the number of users sharing network resources. When a given service is highly demanded, some demands may be unmet due to the limited capacity of the network links. Moreover, for such demands, telecommunication operators should pay penalty costs. To avoid rejecting demands, we can install more capacities in the existing network. In this paper we report experiments on the network capacity design for uncertain demand in telecommunication networks with integer link capacities. We use Poisson demands with bandwidths given by normal or log-normal distribution functions. The expectation function is evaluated using a predetermined set of realizations of the random parameter. We model this problem as a two-stage mixed integer program, which is solved using a stochastic subgradient procedure, the Barahona's volume approach and the Benders decomposition.     

Heuristics for Distribution Network Design in Telecommunication

A distribution network problem arises in a lower level of an hierarchical modeling approach for telecommunication network planning. This paper describes a model and proposes a lagrangian heuristic for designing a distribution network. Our model is a complex extension of a capacitated single commodity network design problem. We are given a network containing a set of sources with maximum available supply, a set of sinks with required demands, and a set of transshipment points. We need to install adequate capacities on the arcs to route the required flow to each sink, that may be an intermediate or a terminal node of an arborescence. Capacity can only be installed in discrete levels, i.e., cables are available only in certain standard capacities. Economies of scale induce the use of a unique higher capacity cable instead of an equivalent set of lower capacity cables to cover the flow requirements of any link. A path from a source to a terminal node requires a lower flow in the measure that we are closer to the terminal node, since many nodes in the path may be intermediate sinks. On the other hand, the reduction of cable capacity levels across any path is inhibited by splicing costs. The objective is to minimize the total cost of the network, given by the sum of the arc capacity (cables) costs plus the splicing costs along the nodes. In addition to the limited supply and the node demand requirements, the model incorporates constraints on the number of cables installed on each edge and the maximum number of splices at each node. The model is a NP-hard combinatorial optimization problem because it is an extension of the Steiner problem in graphs. Moreover, the discrete levels of cable capacity and the need to consider splicing costs increase the complexity of the problem. We include some computational results of the lagrangian heuristics that works well in the practice of computer aided distribution network design.     

A Comparison of Optimal Methods for Local Access Uncapacitated Network Design

We compare some optimal methods addressed to a problem of local access network design. We see this problem arising in telecommunication as a flow extension of the Steiner problem in directed graphs, thus including as particular cases some alternative approaches based on the spanning tree problem. We work with two equivalent flow formulations for the problem, the first referring to a single commodity and the second being a multicommodity flow model. The objective in both cases is the cost minimization of the sum of the fixed (structural) and variable (operational) costs of all the arcs composing an arborescence that links the origin node (switching center) to every demand node. The weak single commodity flow formulation is solved by a branch-and-bound strategy that applies Lagrangian relaxation for computing the bounds. The strong multicommodity flow model is solved by a branch-and-cut algorithm and by Benders decomposition. The use of a linear programming solver to address both the single commodity and the multicommodity models has also been investigated. Our experience suggests that a certain number of these modeling and solution strategies can be applied to the frequently occurring problems where basic optimal solutions to the linear program are automatically integral, so it also solves the combinatorial optimization problem right away. On the other hand, our main conclusion is that a well tailored Benders partitioning approach emerges as a robust method to cope with that fabricated cases where the linear programming relaxation exhibits a gap between the continuous and the integral optimal values.     

B&B Frameworks for the Capacity Expansion of High Speed Telecommunication Networks Under Uncertainty

The purpose of this paper is to investigate branch and bound strategies and the comparison of branch and cut with pure branch and bound approaches on high speed telecommunication network design under uncertainty. We model the problem as a two-stage stochastic program with discrete first-stage (investment) variables. Two formulations of the problem are used. The first one with general integer investment variables and the second one, a variant of the first model, with 0-1 investment variables. We present computational results for three solution approaches: the integer L-shaped (Benders) decomposition, a branch and bound framework and a disjunctive cutting plane method. This work was supported by France Telecom.     

Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem

In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size.     

MC-based Algorithm for a telecommunication network under node and budget constraints

It is an important issue to design some performance indexes in order to measure the performance for a telecommunication network. Network analysis is an available approach to solve the performance problem for a real-life system. We construct a two-commodity stochastic-flow network with unreliable nodes (arcs and nodes all have several possible capacities and may fail) to model the telecommunication network. In which, all types of commodity are transmitted through the same network simultaneously and compete the capacities. This paper defines the system capacity as a 2-tuple vector, and then proposes a performance index, the probability that the upper bound of the system capacity equals a demand vector subject to the budget constraint. An upper boundary point is a vector representing the capacities of arcs and nodes, and is the maximal vector exactly meeting the demand vector. A simple algorithm based on minimal cuts (or named MC-based algorithm) is then presented to generate all upper boundary points in order to evaluate the performance index. The storage and computational time complexity of this algorithm are also analyzed. The performance evaluation for the multicommodity case can be extended easily.     

A cutting plane algorithm for the Capacitated Connected Facility Location Problem

We consider a network design problem that arises in the cost-optimal design of last mile telecommunication networks. It extends the Connected Facility Location problem by introducing capacities on the facilities and links of the networks. It combines aspects of the capacitated network design problem and the single-source capacitated facility location problem. We refer to it as the Capacitated Connected Facility Location Problem. We develop a basic integer programming model based on single-commodity flows. Based on valid inequalities for the capacitated network design problem and the single-source capacitated facility location problem we derive several (new) classes of valid inequalities for the Capacitated Connected Facility Location Problem including cut set inequalities, cover inequalities and combinations thereof. We use them in a branch-and-cut framework and show their applicability and efficacy on a set of real-world instances.     

Tree network design avoiding congestion

We introduce in this paper an optimal method for tree network design avoiding congestion. We see this problem arising in telecommunication and transportation networks as a flow extension of the Steiner problem in directed graphs, thus including as a particular case any alternative approach based on the minimum spanning tree problem. Our multi-commodity formulation is able to cope with the design of centralized computer networks, modern multi-cast multi-party or hub-based transportation trees. The objective in such cases is the minimization of the sum of the fixed (structural) and variable (operational) costs of all the arcs composing an arborescence that links the origin node (switching center, server, station) to every demand node (multi-cast participants, users in general). The non-linear multi-commodity flow model is solved by a generalized Benders decomposition approach.     

Hierarchical approach for survivable network design

A central design challenge facing network planners is how to select a cost-effective network configuration that can provide uninterrupted service despite edge failures. In this paper, we study the Survivable Network Design (SND) problem, a core model underlying the design of such resilient networks that incorporates complex cost and connectivity trade-offs. Given an undirected graph with specified edge costs and (integer) connectivity requirements between pairs of nodes, the SND problem seeks the minimum cost set of edges that interconnects each node pair with at least as many edge-disjoint paths as the connectivity requirement of the nodes. We develop a hierarchical approach for solving the problem that integrates ideas from decomposition, tabu search, randomization, and optimization. The approach decomposes the SND problem into two subproblems, Backbone design and Access design, and uses an iterative multi-stage method for solving the SND problem in a hierarchical fashion. Since both subproblems are NP-hard, we develop effective optimization-based tabu search strategies that balance intensification and diversification to identify near-optimal solutions. To initiate this method, we develop two heuristic procedures that can yield good starting points. We test the combined approach on large-scale SND instances, and empirically assess the quality of the solutions vis-à-vis optimal values or lower bounds. On average, our hierarchical solution approach generates solutions within 2.7% of optimality even for very large problems (that cannot be solved using exact methods), and our results demonstrate that the performance of the method is robust for a variety of problems with different size and connectivity characteristics.     

A branch-and-price approach for the partition coloring problem

This work proposes a new integer programming model for the partition coloring problem and a branch-and-price algorithm to solve it. Experiments are reported for random graphs and instances originating from routing and wavelength assignment problems arising in telecommunication network design. We show that our method largely outperforms previously existing approaches.     

Price-directive decomposition applied to routing in telecommunication networks

The telecommunication network design problem is considered to study the level of transmission network. A heuristic approach is defined to solve the combined routing-grouping problem, where the grouping one is solved by a heuristic approach. The routing problem is defined considering reliability constraints, supplementary circuits demands and a piecewise linear objective function to take into account the influence of the grouping. This last model is solved using a price-directive decomposition method, which has allowed us to solve real networks using an exact method.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

0-1 reformulations of the multicommodity capacitated network design problem

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.     

A genetic algorithm approach on tree-like telecommunication network design problem

In the context of telecommunication networks, the network terminals involve certain constraints that are either related with the performance of the corresponding network or with the availability of some classes of devices. In this paper, we discuss a tree-like telecommunication network design problem with the constraint limiting the number of terminals. First, this problem is formulated as a leaf-constrained minimum spanning tree (lc-MST). Then we develop a tree-based genetic representation to encode the candidate solutions of the lc-MST problem. Compared with the existing heuristic algorithm, the numerical results show the high effectiveness of the proposed GA approach on this problem.     

Minimum concave-cost network flow problems: Applications,complexity, and algorithms

We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.     

A Benders decomposition approach for a distribution network design problem with consolidation and capacity considerations

We develop a model for a strategic freight-forwarding network design problem in which the design decisions involve the locations and capacities of consolidation and deconsolidation centers, and capacities on linehaul linkages as well as the shipment routes from origins to destinations through centers. We devise a solution approach based on Benders decomposition and conduct a computational study that illustrates the efficiency and the effectiveness of the approach.     

Reducing costs of backhaul networks for PCS networks using genetic algorithms

Designing cost-effective telecommunications networks often involves solving several challenging, interdependent combinatorial optimization problems simultaneously. For example, it may be necessary to select a least-cost subset of locations (network nodes) to serve as hubs where traffic is to be aggregated and switched; optimally assign other nodes to these hubs, meaning that the traffic entering the network at these nodes will be routed to the assigned hubs while respecting capacity constraints on the links; and optimally choose the types of links to be used in interconnecting the nodes and hubs based on the capacities and costs associated with each link type. Each of these three combinatorial optimization problems must be solved while taking into account its impacts on the other two. This paper introduces a genetic algorithm (GA) approach that has proved effective in designing networks for carrying personal communications services (PCS) traffic. The key innovation is to represent information about hub locations and their interconnections as two parts of a chromosome, so that solutions to both aspects of the problem evolve in parallel toward a globally optimal solution. This approach allows realistic problems that take 4–10 hours to solve via a more conventional branch-and-bound heuristic to be solved in 30–35 seconds. Applied to a real network design problem provided as a test case by Cox California PCS, the heuristics successfully identified a design 10% less expensive than the best previously known design. Cox California PCS has adopted the heuristic results and plans to incorporate network optimization in its future network designs and requests for proposals.     

A cost allocation problem arising in hub–spoke network systems

This paper studies a cost allocation problem arising from hub–spoke network systems. When a large-scale network is to be constructed jointly by several agents, both the optimal network design and the fair allocation of its cost are essential issues. We formulate this problem as a cooperative game and analyze the core allocation, which is a widely used solution concept. The core of this game is not necessarily non-empty as shown by an example. A reasonable scheme is to allocate the cost proportional to the flow that an agent generates. We show that, if the demand across the system has a block structure and the fixed cost is high, this cost allocation scheme belongs to the core. Numerical experiments are given with real telecommunication traffic data in order to illustrate the usefulness of our analytical findings.     

A GRASP and Path Relinking Heuristic for Rural Road Network Development

This paper presents a model for rural road network design that involves two objectives: maximize all season road connectivity among villages in a region and maximize route efficiency, while allocating a fix budget among a number of possible road projects. The problem is modeled as a bicriterion optimization problem and solved heuristically through a greedy randomized adaptive search procedure (GRASP) in conjunction with a path relinking procedure. The implementation of GRASP and path relinking includes two novel modifications, a new form of reactive GRASP and a new form of path relinking. Overall, the heuristic approach is streamlined through the incorporation of advanced network flow reoptimization techniques. Results indicate that this implementation outperforms both GRASP as well as a straightforward form of GRASP with path relinking. For small problem instances, for which optimality could be verified, this new, modified form of GRASP with path relinking solved all but one known instance optimally.     

Fast, Efficient Equipment Placement Heuristics for Broadband Switched or Internet Router Networks

Planning and designing the next generation of IP router or switched broadband networks seems a daunting challenge considering the many complex, interacting factors affecting the performance and cost of such networks. Generally, this complexity implies that it may not even be clear what constitutes a “good” network design for a particular specification. Different network owners or operators may view the same solution differently, depending on their unique needs and perspectives. Nevertheless, we have observed a core common issue arising in the early stages of network design efforts involving leading-edge broadband switched technologies such as ATM, Frame Relay, and SMDS; or even Internet IP router networks. This core issue can be stated as follows: Given a set of service demands for the various network nodes, where should switching or routing equipment be placed to minimize the Installed First Cost of the network? Note that the specified service demands are usually projections for a future scenario and generally entail significant uncertainty. Despite this uncertainty, we have found that network owners and operators generally feel it is worthwhile to obtain high-level advice on equipment placement with a goal of minimizing Installed First Cost. This paper reports on a heuristic approach we have implemented for this problem that has evolved out of real network design projects. A tool with both a Solution Engine and an intuitive Graphical User Interface has been developed. The approach is highly efficient; for example, the tool can often handle LATA-sized networks in seconds or less on a workstation processor. By using only nodal demands rather than the more complex point-to-point demands usually required in tools of this sort, we have created an approach that is not only highly efficient, but is also a better match to real design projects in which demand data is generally scant and highly uncertain.